A hollow gas turbine blade which can be cooled by cooling air is disclosed in U.S. Pat. No. 5,468,125. The cooling air is blown into cooling chambers, running parallel to the blade axis, of the hollow gas turbine blade. There, passing through the chambers, it cools the hot surface of the gas turbine blade from the inside. The incoming cooling air not yet heated is first of all directed past the leading edge of the gas turbine blade, this leading edge being subjected to especially high temperatures and therefore having to be cooled in an especially efficient manner. After the cooling air has been passed through the blade in such a way as to also cool the other regions of the blade, it leaves the latter at the trailing edge of the blade via holes.